Explicit formulas and recurrence relations for generalized Catalan numbers

In this paper, we present an explicit formula and recurrent relation for generalized Catalan numbers, from which we can give corresponding formulas for Schröder numbers, large and small generalized Catalan numbers for the special cases of our results.

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[1] Bourbaki N. Functions of a Real Variable, Elementary Theory. Translated from the 1976 French Original by Philip Spain. Elements of Mathematics (Berlin). Springer, Berlin: 2004.
[2] Brualdi RA. Introductory Combinatorics. Fifth edition, Pearson Prentice Hall, Upper Saddle River, NJ: 2010.
[3] Cao HQ, Pan H. A Stern-type congruence for the Schröder numbers. Discrete Mathematics 2017; 340: 708-712.
[4] Chen Z, Pan H. Notes on the q−colored Motzkin numbers and Schröder numbers. Journal of Difference Equations and Applications 2017; 23 (6): 1133-1141.
[5] Comtet L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition. D. Reidel Publishing Co., Dordrecht: 1974.
[6] He TX. Parametric Catalan numbers and Catalan triangles. Linear Algebra and its Applications 2013; 438: 1467- 1484.
[7] Koparal S, Ömür N. On congruences involving the generalized Catalan numbers and harmonic numbers. Bulletin of the Korean Mathematical Society 2019; 56: 649-658.
[8] Koparal S, Ömür N. Some congruences involving Catalan, Pell and Fibonacci numbers. Mathematica Montisnigri 2020; 48: 10-18.
[9] Koshy T. Catalan Numbers with Applications. Oxford University Press, Oxford: 2009.
[10] Kruchinin DV, Kruchinin VV. Application of a composition of generating functions for obtaining explicit formulas of polynomials. Journal of Mathematical Analysis and Applications 2013; 404: 161-171.
[11] Liu JC. A supercongruence involving Delannoy numbers and Schröder numbers. Journal of Number Theory 2016; 168: 117-127.
[12] Liu JC. Proof of Sun’s conjectural supercongruence involving Catalan numbers. Electronic Research Archive 2020; 28 (2): 1023-1030.
[13] Liu JC. On a congruence involving q−Catalan numbers. Comptes Rendus Mathématique 2020; 358: 211-215
[14] Liu JC, Huang ZY. A truncated identity of Euler and related q−congruences. Bulletin of the Australian Mathematical Society 2020; 102: 353-359.
[15] Liu JC, Li L, Wang SD. Some congruences on Delannoy numbers and Schröder numbers. International Journal of Number Theory 2018; 14: 2035-2041.
[16] Natalini P, Ricci PE. Higher order Bell polynomials and the relevant integer sequences. Applicable Analysis and Discrete Mathematics 2017; 11: 327-339.
[17] Qi F. Parametric integrals, the Catalan numbers, and the beta function. Elemente Der Mathematik 2017; 72: 103-110.
[18] Qi F, Akkurt A, Yildirim H. Catalan numbers, k-gamma and k-beta functions, and parametric integrals. Journal of Computational Analysis and Applications 2018; 25: 1036-1042.
[19] Qi F, Guo BN. Integral representations of the Catalan numbers and their applications. Mathematics 2017; 5: 40.
[20] Qi F, Mahmoud M, Shi XT, Liu FF. Some properties of the Catalan–Qi function related to the Catalan numbers. SpringerPlus 2016; 5: 1126.
[21] Qi F, Shi XT, Guo BN. Two explicit formulas of the Schröder numbers. Integers 2016; 16: A23.
[22] Qi F, Shi XT, Mahmoud M, Liu FF. The Catalan numbers: a generalization, an exponential representation, and some properties. Journal of Computational Analysis and Applications 2017; 23: 937-944.
[23] Qi F, Zou Q, Guo BN. The inverse of a triangular matrix and several identities of the Catalan numbers. Applicable Analysis and Discrete Mathematics 2019; 13: 518-541.
[24] Qi F, Shi XT, Guo BN. Some properties of the Schröder numbers. Indian Journal of Pure and Applied Mathematics 2016; 47 (4): 717-732.
[25] Qi F, Guo BN. Some explicit and recursive formulas of the large and little Schröder numbers. Arab Journal of Mathematical Sciences 2017; 23: 141-147.
[26] Qi F, Guo BN. Several explicit and recursive formulas for generalized Motzkin numbers. AIMS Mathematics 2020; 5 (2): 1333-1345.
[27] Qi F, Guo BN. Explicit and recursive formulas, integral representations, and properties of the large Schröder numbers. Kragujevac Journal of Mathematics 2017; 41 (1): 121-141.
[28] Qi F. Derivatives of tangent function and tangent numbers. Applied Mathematics and Computation 2015; 268: 844-858.
[29] Qi F, Zheng MM. Explicit expressions for a family of the Bell polynomials and applications. Applied Mathematics and Computation 2015; 258: 597-607.
[30] Qi F, Guo BN. Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials. Mediterranean Journal of Mathematics 2017; 14 (3): Article 140.
[31] Qi F, Lim D, Guo BN. Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations. Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales. Seria A. Matematicas RACSAM 2019; 113: 1-9.
[32] Qi F. An Explicit Formula for the Bell Numbers in Terms of the Lah and Stirling Numbers. Mediterranean Journal of Mathematics 2016; 13: 2795-2800.
[33] Stanley RP. Enumerative Combinatorics (Vol. 2). Cambridge Studieds in Advanced Mathematics; Cambridge University Press: Cambridge, UK: 1997.
[34] Sun ZW. On Delannoy numbers and Schröder numbers. Journal of Number Theory 2011; 131: 2387-2397.
[35] Sun ZW. Arithmetic properties of Delannoy numbers and Schröder numbers. Journal of Number Theory 2018; 183: 146-171.
[36] Withers CS, Nadarajah S. Multivariate Bell polynomials. International Journal of Computer Mathematics 2010; 87: 2607-2611.
[37] Withers CS, Nadarajah S. Multivariate Bell polynomials, series, chain rules, moments and inversion. Utilitas Mathematica 2010; 83: 133-140.
[38] Withers CS, Nadarajah S. Multivariate Bell polynomials and their applications to powers and fractionary iterates of vector power series and to partial derivatives of composite vector functions. Applied Mathematics and Computation 2008; 206: 997-1004.